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Annual Conference on Magnetism and Magnetic Materials (MMM)

Electron Correlation in a Narrow Band Metal a New Method of Calculation

Samuel P. Bowen

Published Jan 1, 1974• 0 citations
Publisher

Abstract

A generalization of Feenberg's Perturbation Theory is shown to give exact expressions for the temperature dependent retarded greens functions for a many body problem. The theorem establishing this relationship is sketched and the result briefly applied to the Hubbard model.
DOI
10.1063/1.3141799

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To investigate the effects of electron correlation in narrow energy bands, a general formula of static spin susceptibility for interacting electron systems is derived. It is expressed in terms of the one-particle Green's function and its derivative with respect to the average number of the opposite spin electrons, the existence of the latter being the characteristic of electron correlation. The formula is applied to the case of Hubbard's improved approximation (Hubbard III), and gross feature of both the correlation and temperature dependences of the susceptibility is investigated in the insulating and metallic regions. The correlation in the metallic region is reduced for the Hubbard III case so that there occurs no ferromagnetism in a half-filled band.
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An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's TTT-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
Beyond the usual many body perturbation theory, with the eigenfunctional theory, we directly calculate the correlation function produced by the Coulomb interaction of electrons in the equation of one-electron Green function, and give the general expression of the non-local effective interaction potential in a Hartree-type potential, which is absent in all previous many body perturbation theories. This effective interaction potential originates from the quantum many body effect of the system, and it cannot be obtained directly by the usual perturbation expression approach. Moreover, using theoretical models, we demonstrate that this effective interaction potential can be used to characterize the electron correlation strength of the system.